The footing that hangs from its column

A footing strip sits on the ground as a beam held everywhere, and the ground under it is not elastic. Each slice of it yields along its loading stiffness when it is pressed further than ever before, comes back along a stiffer line when the pressure comes off, keeps a permanent set, and has to be pressed past its old maximum before it yields again. The footing the storms press down rocked such a strip with storms that reversed evenly and found that it pressed a shallow bowl in a storm or two and then rocked inside it without pressing further. The footing a prevailing wind leans pushed harder one way than the other and found a lean — towards the harder push — that stopped within three storms and grew afterwards only when a storm beat every one before it.

In both, the column load was the same through the storm, and large enough that the strip was always pressing down somewhere. A footing under a column of a bracing bay is different. The wind’s overturning moment on the bay is carried by its columns as a couple — the leeward column pressed down, the windward one pulled up — so the column’s load swings with the storm: heavier when the wind pushes one way, lighter, or in tension, when it swings the other. If the pull is larger than the dead load and the footing’s own weight, the footing is lifted clear of the ground and hangs from the column for part of every storm.

The question the prevailing wind left is whether such a footing leans the way a rocked one does, or whether being lifted clear of its bowl in every reverse storm lets it come down somewhere new each time, and walk.

A column pressed one way and pulled the other #

Take the strip of the earlier essays — 8 m long, a bending stiffness of $540\ \mathrm{MN\,m^2}$, on a bed of 50 MN/m² that unloads and reloads four times as stiffly as it first loads — under a column carrying 1,000 kN of dead load. Give the strip a weight of its own, 40 kN a metre, which a footing designed to resist uplift must have: 320 kN in all, holding it down. A bracing column’s base carries little moment of its own, so let each storm reverse a moment of 300 kN·m fully, and let it carry the column’s load with it, $P\,(1 + s\,m/M)$, the swing $s$ being the share of the dead load added at the full push and taken away at the full swing back.

The three storms trace straight lines through the same point, the dead load at no moment. With no swing the load is constant and the strip is the rocked strip of the earlier essays at a third of their moment. At 100 per cent swing the column is pressed with twice its dead load at the push and carries nothing at the swing back, so the strip is held down by its own weight alone. At 150 per cent the column pulls 500 kN at the swing back, more than the strip weighs, and for the last part of every storm — past 264 kN·m of swing back — the strip is hanging from the column with no contact at all.

Two things make that case worth asking about. While the strip hangs, nothing presses the ground: whatever memory the ground has, it keeps unchanged. And when the storm eases and the strip comes back down, it lands on a bowl it pressed earlier, which has been waiting for it. Whether it lands where it left, or somewhere slightly different that then becomes the new starting point, is what would decide a walk.

The lean, storm by storm #

Run eight storms of 300 kN·m each, at five swings from none to 200 per cent, and read the column’s tilt at rest under its dead load after each.

The storms that do not swing the load leave the footing upright, as the evenly rocked footing was: three thousandths of a milliradian after eight storms, which is nothing. Every storm that swings the load leaves a lean of about a tenth of a milliradian, towards the end the heavier half of the storm presses — the column’s push and its extra load arrive together, so the ground under that end is pressed further than the ground under the other.

The lean behaves exactly as the prevailing wind’s did. It arrives in the first storm, eases slightly in the second and third as the other end’s ground catches up, and then stops. By the third storm every curve is flat to the third decimal. The two swings that hang the strip clear in every storm — 150 and 200 per cent — are not different in kind from the ones that do not: they lean a little more than the 100 per cent swing and a little less than the 50, and they stop at the same time.

So a footing that is alternately pressed and pulled leans the same way a footing in a prevailing wind does. The swing of the load does what the prevailing wind’s lopsided moment did: it makes one half of each storm press harder than the other. And being lifted clear makes no difference.

Where the footing hangs, nothing is remembered #

The reason is visible in one storm. Follow the column’s rotation through the eighth storm at 150 per cent swing, against the storm’s moment.

On the ground the rotation follows one line out and back, because by the eighth storm the ground under every slice is on its stiff reloading branch: nothing is pressed past its old maximum, so nothing is added to the memory, and the path is elastic. The line steepens as the storm swings back and the load lightens: a light footing rocks more for the same moment, because less of its length stays in contact. Then the strip leaves the ground altogether.

While it hangs, the model has nothing to say about where the strip is: its position is the column’s, held by the frame above it. What the model does say is that the ground’s memory is untouched, because nothing touches it. When the storm turns and the load comes back, the strip settles onto the reloading branch of springs whose sets and maxima are exactly what they were when it left, and the only position in which those springs balance the dead load is the one it had before. The bowl does the locating. A footing whose ground remembers comes down where its ground expects it.

This is the base that lets go and closes again one storey lower. A column base lifted off its bed by a storm’s moment closes back onto the same bed, and what decides its future is what the lift-off did to its bolts. Here the lift-off does nothing to the ground, so nothing changes.

The footing sinks to its heaviest press #

What the swing does do is visible in the settlement rather than the tilt.

The heavier half of each storm presses the footing with the dead load plus the swing, and the ground keeps that press’s set. With no swing the settlement at rest is 5.39 mm, a little more than the 5.05 the dead load alone makes on new ground, because the storms’ moment pressed each end further than the dead load does. With the load swinging by 100 per cent the column has pressed with 2,000 kN in every storm and the settlement at rest is 7.87 mm; at 200 per cent, 3,000 kN and 10.66 mm. The settlement doubles across the range while the lean hardly changes.

At rest under its dead load, the footing stands above the settlement its heaviest press made — 7.87 against 9.31 at 100 per cent — because the ground came back along its stiff branch when the press came off. That is the load it can carry once in the soil: the ground was pressed past its old state once, kept the set, and every press since has been elastic within it. The pull of the lighter half takes nothing back. A spring unloaded to its set opens a gap rather than pulling the strip up, and a strip hanging clear does not lift the ground with it.

A deeper bowl, barely tilted #

Read the ground’s permanent set along the strip after six storms.

The set is a bowl under the whole strip, deepest under the column and shallower towards the ends, as the rocked footing’s was. With the load swinging it is two to three times as deep, and it is hardly tilted at all: at 100 per cent the two ends’ sets are equal to the hundredth of a millimetre. The prevailing wind’s bowl was 0.30 mm deep at one end and 0.91 at the other.

The difference is in what drives the lean. A prevailing wind pushes one way harder, so the end it pushes towards is pressed further than the other and the bowl tilts. A swinging load presses harder in one half of the storm but presses the whole strip down as it does so; the extra load is central, and it deepens both ends nearly alike. The small tilt left is the moment’s, made asymmetric by the load that comes with it. The column’s lean, a tenth of a milliradian, is 0.8 mm across 8 m, against sets that differ between the two ends by under a tenth of a millimetre: the lean is not the tilt of the bowl’s rim but how the strip comes to rest on the bowl as a whole.

The swing deepens the footing more than it leans it #

Sweep the swing from none to 200 per cent, at two storm moments.

The tilt jumps from nothing to about its final value as soon as the load swings at all, and then wanders within a fifth of it whatever the swing. It is set mainly by the storm’s moment — 0.1 milliradians at 300 kN·m, 0.2 at 600 — and only a little by how much the load swings with it. A designer worried about a bracing column’s footing leaning in a storm needs the moment and the ground’s memory, and can leave the swing aside; a designer worried about its settlement needs the swing above all, because the heaviest press decides how deep the footing will stand for the rest of its life.

By hand: when it hangs, and how deep it sinks #

Both numbers that matter here can be had without the strip. The footing hangs when the column’s pull passes its own weight: with the load $P(1 + s\,m/M)$ and a strip weighing $W$, at a moment on the swing back of

$$m_{hang} = \frac{(P + W)\,M}{P\,s},$$

which for 1,000 kN, 320 kN and 300 kN·m is 396 kN·m at 100 per cent swing — beyond the storm, so it never hangs — and 264 at 150 per cent. It hangs at all only if $s > 1 + W/P$, here 1.32. The holding-down check a designer makes is the same inequality with a factor on it, and it says nothing about what the ground does in the meantime.

The settlement at rest is the heaviest press’s settlement, less what comes back when the press is lifted. On a bed that unloads $r$ times as stiffly as it loads, a press that settles the footing $w_{max}$ leaves a set of $w_{max}(1 - 1/r)$, and the dead load then stands on the stiff branch above it, a further $w_{dead}/r$ down. With $r = 4$, the press of 2,000 kN at 100 per cent swing settles the footing about 9.3 mm, leaving a set of 7.0; the dead load’s 5.05 mm on new ground becomes 1.3 on the stiff branch; and together they make 8.2 mm, against the 7.87 the strip finds once the storms’ rocking has spread the press along it. The estimate is a little high, and it is the right way round to be wrong.

Neither calculation needs the lean, and the lean needs neither of them: it comes from the storm’s moment acting with a load that differs between the two halves of the storm, which is the prevailing wind’s mechanism at a different scale.

What the holding-down weight is for #

The strip’s weight was chosen here so that a swing of 150 per cent would just lift it, and it is worth saying what the weight is doing when the swing is smaller. A footing designed against uplift is made heavy enough that the column’s pull never lifts it, and the ends the ground was holding down found that a strip’s ends can lift off the ground under a moment long before the whole strip does. Under a swinging load the two happen together: the lighter the load in the swing back, the less of the strip stays in contact and the more it rocks, until at the hang there is no contact at all.

None of that moves the footing for good, for the same reason the hang does not: the parts of the strip that leave the ground stop pressing it, and the ground keeps its memory. What the weight protects is the column’s connection to the footing, which carries the whole pull while the footing hangs — the holding-down bolts that lose their preload once in a storm and then settle — and the ground beam or slab that would otherwise have to hold the footing in place. The ground does not need it.

What the strip leaves out #

The ground is a bed of independent springs, each with its own memory and no knowledge of its neighbours. Real soil spreads a press sideways, so the set under one end deepens a little under the other, which makes a bowl rounder than the springs say and leaves the conclusions about leaning unchanged. The springs never slide: a footing that hangs clear and comes back down is not displaced horizontally, because there is nothing in the model to displace it, and a real bracing bay does push its footing sideways, by the bay’s shear, as it lifts and settles. A footing that walked would walk that way, by sliding on its bed in the moments when it is lightest — which friction under its own weight resists, and which a ground beam tying it to its neighbours prevents outright.

And the storms are identical. Under a record of storms of different sizes, as the prevailing wind’s fifty years showed, the lean grows only in the years when a storm beats every one before it. A footing whose load swings would behave the same way, with a deeper bowl after each record and a lean that follows the record moment rather than the record load.

A footing held by its bowl #

A footing pulled clear of the ground for part of every storm sounds like a footing that has lost what was holding it. It has lost its contact for a moment, and nothing else. The ground’s memory is a set and a maximum at every point, and only a press past the maximum changes it; a footing that is hanging presses nothing, and the memory waits. When the load returns, the footing settles onto springs that are exactly as it left them, and the bowl they make puts it back where it was.

So a bracing column’s footing leans once, by about what the storm’s moment would lean a footing whose load did not swing, and then stops; it sinks once, by what its heaviest press makes; and it does not walk. What the swing of the load buys is depth. A footing designed for the dead load and checked for overturning is checked for the two things that do not happen, and the settlement that does — twice the dead load’s, set by the storm’s heaviest press — is the number to design against.

Still open: the two footings of one bay #

Every footing here is alone, with its column’s load prescribed by the storm. The two columns of a bracing bay stand on two footings, and the bay’s overturning is shared between them through the frame above: the leeward footing is pressed as the windward one is pulled. As the leeward footing settles into its deeper bowl, the bay rotates, and a frame that rotates sheds some of its overturning to the other bays and its own footings to the ground differently from a frame on rigid supports. Whether two footings that settle by different amounts in alternating storms tilt the bay by the difference, whether the bay’s rotation then changes how the storm’s load is shared between them, and whether the pair of bowls lean the bay further than either footing leans its column, is what the frame between them would ask of the strips that were solved alone here.